Question

A vegetable distributor knows that during the month of August, the weights of its tomatoes are normally distributed with a mean of 0.63 lb and a standard deviation of 0.15 lb. (See Example 2 in this section.)

(a) What percent of the tomatoes weigh less than 0.78 lb?
A. 68%
B. 84%
C. 93%
D. 97%

(b) In a shipment of 8,000 tomatoes, how many tomatoes can be expected to weigh more than 0.33 lb?
A. 1,440 tomatoes
B. 2,720 tomatoes
C. 3,200 tomatoes
D. 4,080 tomatoes

(c) In a shipment of 3,500 tomatoes, how many tomatoes can be expected to weigh from 0.33 lb to 0.93 lb?
A. 1,225 tomatoes
B. 1,750 tomatoes
C. 2,450 tomatoes
D. 2,850 tomatoes

Answer 1

a) Approximately 84% of the tomatoes weigh less than 0.78 lb. b) Approximately 2.28% of the tomatoes can be expected to weigh more than 0.33 lb in a shipment of 8,000 tomatoes. c) Approximately 95.44% of the tomatoes can be expected to weigh from 0.33 lb to 0.93 lb in a shipment of 3,500 tomatoes.

Step-by-step explanation:

a) To find the percentage of tomatoes that weigh less than 0.78 lb, we need to calculate the z-score first. The z-score formula is:
z = (x - mean) / standard deviation
Substituting the given values:
z = (0.78 - 0.63) / 0.15 = 1
We then look up the corresponding cumulative probability for a z-score of 1, which is approximately 0.8413 or 84.13%. Therefore, approximately 84% of the tomatoes weigh less than 0.78 lb.

b) To find the number of tomatoes expected to weigh more than 0.33 lb in a shipment of 8,000 tomatoes, we can use the z-score again. The z-score formula remains the same:
z = (x - mean) / standard deviation
Substituting the given values:
z = (0.33 - 0.63) / 0.15 = -2
We then look up the corresponding cumulative probability for a z-score of -2, which is approximately 0.0228 or 2.28%. Therefore, approximately 2.28% of the tomatoes can be expected to weigh more than 0.33 lb.

In an 8,000 tomato shipment, we can expect approximately 2.28% of 8,000, which is 182.4 tomatoes, to weigh more than 0.33 lb.

c) To find the number of tomatoes expected to weigh from 0.33 lb to 0.93 lb in a shipment of 3,500 tomatoes, we need to find the cumulative probability for both weights separately and subtract them. First, we find the z-scores for both weights:
z1 = (0.33 - 0.63) / 0.15 = -2
z2 = (0.93 - 0.63) / 0.15 = 2
We then look up the corresponding cumulative probabilities for these z-scores:
p1 = 0.0228 (from part b)
p2 = 0.9772 (1 - p1)
Subtracting the cumulative probabilities:
p2 - p1 = 0.9772 - 0.0228 = 0.9544 or 95.44%
Therefore, approximately 95.44% of the tomatoes can be expected to weigh from 0.33 lb to 0.93 lb in a shipment of 3,500 tomatoes, which is approximately 3,500 * 0.9544 = 3,344 tomatoes.